See below. Hard to tell although a muscle strain or pull is likely a partial tearing of the muscle fibers. This can be small and microscopic level or be large and obvious on mri. It is not unsual for there to be trouble taking a deep breath with this. If it is not improving after a week or so, you should see a doctor and get checked out. Sometimes physical therapy can be helpful for this.

Tear vs pull no diff. Most of the time, those words mean the same thing, an injury to a muscle which will heal. Based on your description that what it sounds like, a muscle injury that will heal. If the tendon were ruptured, surgery is a consideration. If the lat tendon ruptured, there would be lots of bruising & deformity in the armpit. This is unlikely. The most common risk factor for a lat tendon rupture is steroids.